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21: 10.6 Recurrence Relations and Derivatives

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§10.6(i) Recurrence Relations

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$H_0^{(1)}{}_{}{}^{\prime }\left(z\right)=-H_1^{(1)}\left(z\right),$ $H_0^{(2)}{}_{}{}^{\prime }\left(z\right)=-H_1^{(2)}\left(z\right).$

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§10.6(ii) Derivatives

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10.6.7 $${𝒞}_{\nu }^{(k)}\left(z\right)=\frac{1}{2^k}\sum _{n=0}^k{(-1)}^n\left(\genfrac{}{}{0.0pt}{}{k}{n}\right){𝒞}_{\nu -k+2n}\left(z\right).$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${𝒞}_{\nu }\left(z\right)$: cylinder function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.31

Referenced by:

§10.6(ii)

Permalink:

http://dlmf.nist.gov/10.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.6(ii), §10.6 and Ch.10

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22: 10.53 Power Series

… ►For ${𝗁}_n^{(1)}\left(z\right)$ and ${𝗁}_n^{(2)}\left(z\right)$ combine (10.47.10), (10.53.1), and (10.53.2). …

23: 10.57 Uniform Asymptotic Expansions for Large Order

… ►Asymptotic expansions for ${𝗃}_n\left((n+\frac{1}{2})z\right)$, ${𝗒}_n\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(1)}\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(2)}\left((n+\frac{1}{2})z\right)$, ${𝗂}_n^{(1)}\left((n+\frac{1}{2})z\right)$, and ${𝗄}_n\left((n+\frac{1}{2})z\right)$ as $n\to \mathrm{\infty }$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). …

24: 10.9 Integral Representations

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Mehler–Sonine and Related Integrals

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Schläfli–Sommerfeld Integrals

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Hankel’s Integrals

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§10.9(iv) Compendia

►For collections of integral representations of Bessel and Hankel functions see Erdélyi et al. (1953b, §§7.3 and 7.12), Erdélyi et al. (1954a, pp. 43–48, 51–60, 99–105, 108–115, 123–124, 272–276, and 356–357), Gröbner and Hofreiter (1950, pp. 189–192), Marichev (1983, pp. 191–192 and 196–210), Magnus et al. (1966, §3.6), and Watson (1944, Chapter 6).

25: 11.2 Definitions

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11.2.13 $$w={𝐇}_{\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{H}_{\nu }^{(1)}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), ${𝐇}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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11.2.14 $$w={𝐇}_{\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{H}_{\nu }^{(2)}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), ${𝐇}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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11.2.15 $$w={𝐊}_{\nu }\left(z\right)+A{H}_{\nu }^{(1)}\left(z\right)+B{H}_{\nu }^{(2)}\left(z\right).$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), ${𝐊}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii), §11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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26: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

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10.20.6 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu z\right)\\ H_{\nu }^{(2)}\left(\nu z\right)\end{array}\}\sim 2{\mathrm{e}}^{\mp \pi \mathrm{i}/3}{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.37

Referenced by:

§10.20(iii), §10.41(v)

Permalink:

http://dlmf.nist.gov/10.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

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10.20.9 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu z\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu z\right)\end{array}\}\sim \frac{4{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}}{z}{\left(\frac{1-z^2}{4\zeta }\right)}^{\frac{1}{4}}\left(\frac{{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{4}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{C_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{2}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{D_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $C_k(\zeta )$: coefficients and $D_k(\zeta )$: coefficients

A&S Ref:

9.3.45

Referenced by:

§10.20(ii)

Permalink:

http://dlmf.nist.gov/10.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

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§10.20(iii) Double Asymptotic Properties

►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).

27: 10.42 Zeros

… ►Properties of the zeros of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ may be deduced from those of $J_{\nu }\left(z\right)$ and $H_{\nu }^{(1)}\left(z\right)$, respectively, by application of the transformations (10.27.6) and (10.27.8). …

28: 10.73 Physical Applications

… ►The functions ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗁}_n^{(1)}\left(x\right)$, and ${𝗁}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): …With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. …

29: 15.14 Integrals

… ►Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …

30: 10.51 Recurrence Relations and Derivatives

… ►Let $f_n(z)$ denote any of ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, or ${𝗁}_n^{(2)}\left(z\right)$. …